Edit /etc/ssh/sshd_config and set the parameter PAMAuthenticationViaKBDInt to yes

# To disable tunneled clear text passwords, change PasswordAuthentication to no.
PasswordAuthentication yes

# Use PAM via keyboard interactive method for authentication.
# Depending on the setup of pam.conf(4) this may allow tunneled clear text
# passwords even when PasswordAuthentication is set to no. This is dependent
# on what the individual modules request and is out of the control of sshd
# or the protocol.
#PAMAuthenticationViaKBDInt no
PAMAuthenticationViaKBDInt yes

After you set up the public key authentication in any Unix server, if it keeps asking for your password even after you put your public key in the ~/.ssh/authorized_keys file you need to check the file permissions.

To fix this problem, check if the owner and the file permission is set right on:
Home directory (700)
.ssh directory (700)
authorized_keys file (600)